A345315 a(n) = Sum_{d|n} d^[Omega(d) = 2], where [ ] is the Iverson bracket.

1, 2, 2, 6, 2, 9, 2, 7, 11, 13, 2, 14, 2, 17, 18, 8, 2, 19, 2, 18, 24, 25, 2, 16, 27, 29, 12, 22, 2, 36, 2, 9, 36, 37, 38, 25, 2, 41, 42, 20, 2, 46, 2, 30, 28, 49, 2, 18, 51, 39, 54, 34, 2, 21, 58, 24, 60, 61, 2, 43, 2, 65, 34, 10, 68, 66, 2, 42, 72, 64, 2, 28, 2, 77, 44, 46, 80

Offset: 1

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

• nonn

Comments

For each divisor d of n, add d if d is semiprime, otherwise add 1. For example, the divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24, and the only semiprime divisors of 24 are 4 and 6, so a(24) = 1 + 1 + 1 + 4 + 6 + 1 + 1 + 1 = 16.

Inverse Möbius transform of 1 + (n - 1)*c(n), where c = A064911. - Wesley Ivan Hurt, Jul 22 2025

Examples

			a(12) = Sum_{d|12} d^[Omega(d) = 2] = 1^0 + 2^0 + 3^0 + 4^1 + 6^1 + 12^0 = 14.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000005 (tau), A001222 (Omega), A001358 (semiprimes), A064911, A076290, A086971.

Programs

Mathematica

Table[Sum[k^KroneckerDelta[PrimeOmega[k], 2] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

PARI

a(n) = sumdiv(n, d, if (bigomega(d)==2, d, 1)); \\ Michel Marcus, Jun 13 2021

Formula

From Wesley Ivan Hurt, Jul 22 2025: (Start)

a(p^k) = k + p^2 for p prime and k >= 2, else k + 1 if k = 0 or 1.

a(n) = tau(n) + Sum_{d|n} (d - 1) * c(d), where c = A064911.

a(n) = A076290(n) + A000005(n) - A086971(n). (End)